358 Infinite Divisibility of Finite Matter. 



parts; which is proved by the fact that we can move over it. 

 Whereas, if the least possible part of any line did contaii;, and 

 could be divided into, an infinite number of infinitely divisible 

 parts, and we were to move, as we necessarily must, first over 

 half these parts, then over half the remainder, and so on in this 

 ratio, we conld never, to all eternity, move over all the parts 

 of the least possible part; for there would always be half of a 

 resnainder left- But the fact is, there is no absolute distance 

 between the begiiKiing nnd the e.)d of the least pos.-ible part, 

 for such a part itself is as indivisible as the centre of a circle. 

 The impossibility involved sn the proposition is, that the last 

 remaining part cannot be divided into halves. Therefore, 

 when in moving over any finite line, we have moved over all 

 the least possible parts of it but one, if we move any further, 

 we must move over that one, which brings us to the end of 

 the line. 



N. B. The reason why the lines of the Asymtot - will 

 never come in contact, is because they approach each ither 

 so slowly. According to the nature of their approaching, it 

 must take to all eternity for them to touch each other Sup- 

 pose that one should move so slowly that it would take to all 

 eternity to go from his house to his office, how long would he 

 be in going half the distance? Would he be as long as he 

 would be in going the whole distance? 



How to divide a Finite Material Lme into indivisible parts. 



First describe a circle, and make its diameter the line to be 

 divided. The centre of a circle is indivisible, and so is the 

 line of its periphery, taken lengthwise ; the diameter, or line 

 to be divided, therefore, has three indivisible points: its cen- 

 tre, which is the centre of its circle, and its points of union, 

 with its periphery. Each of the semi-diameters will be di- 

 vided into as many parts as you describe circles within its own 

 circle. Now if you describe its own circle perfectly full of 

 Jesser circles, that is, if you make the indivisible periphery 

 of the first inner circle to come into actual contact with the 

 indivisible periphery of its own circle,* and the indivisible 

 periphery of the second inner circle into actual contact with 



* Viz. of thp oircl*^ of which the sriven line is the diameter. 



